1 Photons: X-rays, γ- rays; electrons, positrons Lecture 2 Shell structure of the atoms. Notion of the cross section of the interaction

2 The electromagnetic spectrum

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4 Each photon has a wavelength λ inversely proportional to its energy E, X-ray: Particle-wave duality. For interaction with aperiodic systems X-rays behave mostly like particles. h is the Planck’s constant and c is the speed of light.

5 Electronic structure of atoms Energy levels of the atom consisting of a nucleus of charge Z and a single electron: Here ℏ =h/2π, n principle quantum number. For given n angular momentum quantum number, l, could be between 0 and n-1. “The rotation axis” can be oriented in 2l+1 ways: The electron states are labeled by n and l. l=0,1,2,3,4,5,6 corresponds to s,p,d,f,g,h. Principal number is written as a number 5d state is n=5, l=2 state.

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7 Energy levels for multielectron atoms Prime effect: Spin statistics - electrons have spin 1/2 - fermions. Cannot be in identical states. For given space wave function - 2 states - electron with spin oriented in two opposite directions: S=+1/2 and S=- 1/2. If we neglect the electron-electron interaction - effectively screening of the Coulomb field of the nucleus by the cloud of the electrons, we have the hydrogen type QM problem with condition that two electrons should not be in the same quantum state. Account of the interaction: electronic levels are functions of both n and l, though the difference is slight. However it leads to reordering of the filling of the electron states.

8 Schematic diagram showing electron structure for hydrogen, helium, lithium, carbon and neon.

9 Ground state atoms: all electrons are in the lowest states Excited atom, one or more electrons are in excited state. Takes 10 s to drop to a vacant state. Emission of photons - fluorescence or X-rays. Shell structure : Electrons with same n have approximately the same energy. Shells characterized by the value of the principle number n. n=1,2,3,4,5, ----> K,L,M,N,.. K-shell 2 electrons, L -shell =2(1+3)=8 electrons. Subshell - the same l,n. It (sub)shell is filled - closed (sub)shell. - Important in chemistry of atom interaction. -7

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11 The  electron  configuration for  say  iron  indicates  an argon  electronic  core  see argon  plus  six  d  electrons and  two  s  electrons  The ionization  energy  is  the  least energy  necessary  to  remove to  in  nity  one  electron  from an  atom  of  the  element  11

12 Charact. X-ray LMNOPK L M N O0.08 Characteristic X-rays of Tungsten

13 Coherent scattering Photoelectric effect Compton scattering Pair production Photodisintegration Basic interactions between X-rays and atoms.

14 Attenuation is statistical in nature. Atoms are randomly distributed in the media. Quantum mechanics - the uncertainty principle does not allow too accurate transverse localization of the beam. Need probabilistic language.

15 Let us first consider propagation of a particle through a homogeneous media. Number of particles which entered is N. Number which survived when going the distance L: N(L); N(0)=N. Interactions in different points are independent. Hence probability to interact between points x and x+dx for any of the particles which reached x should be independent of x. Hence N(x)-N(x+dx)=N(x)μdx.

16 Physical meaning of μ, attenuation coefficient Half-value layer: reduces initial flux by a factor of 2. L= ln 2 / μ Average path before the interaction? Probability to reach point x, P(x) ~ exp(-μ x). Exponential decrease → doubling the thickness from L to 2L reduces flux by a factor of 4. However this refers to only incident particles. Photons produce electrons, photons,... cascade propagates further than initial beam - will discuss later. λ=1/μ - mean free path

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19 Graphs of linear attenuation coefficient  The linear attenuation coefficient  can be obtained from tables, or from automated databases such as the NIST database:

20 How μ depends on the properties of media? Incoming particle interacts independently with individual atoms. Probability of interaction depends on number of atoms (mass per unit area) along the path.

21 Dimension of σ ⇒ n=2 where σ is the absorption cross section of interaction ( I will come back to discussion of this quantity next week). It has dimension of length 2 = area ⇒ Since μ,λ are determined by local properties of the media dependence on the density, ρ, should linear:

22 Measurement of σ requires a thin target, narrow beam and detector far away at zero angle. Nice for particle & nuclear physics - unrealistic for medical applications.

23 ⇒ Convenient to tabulate μ/ρ It has the meaning of absorption per unit mass (1 g) of the material in the unit transverse area. To get μ from it, you have to multiply by ρ.

24 The tables and plots of the X-ray mass coefficients are available from the NIST web page tml denotes energy loss

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26 Per gram of material Hydrogen is less effective absorber than oxygen for low energies. However at higher energies situation is reversed.

27 Beam of photons with max energies 400 KeV when going through lead filters out say 50 KeV photons. If the beam max energy is 20 MeV, a lead filter would shift the spectrum to 2-3 MeV. A filter made of light elements will enhance hard component of the spectrum.

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29 Photon total cross sections as a function of energy for scattering off carbon and lead showing contributions of different processes:

30 σ depends on the composition of the target, on the energy of the beam. Absorber removes from the beam the component which interacts stronger with the media. Deviations from the exponential dependence on the distance. It is easy to include dependence of ρ on longitudinal coordinate z: 30

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